Nonlocal equations for coupled heat and mass transport are developed within the Green-Kubo formalism. Nonlocal thermal transport in Lennard-Jones solids is computed to establish the existence of semi-ballistic transport. Deviations from the diffusive theory are shown by comparing the Fourier transform of the response function from the nonlocal theory to that of the diffusive one. It is shown that the deviations from the local theory correspond to acoustic phonons, whose frequency dependence gives rise to the observed deviations from the local theory.